Four Properties

First of all, we observe that this wave packet consists of a real amplitude, a 'sinc' function, multiplied by an exponential phase factor, which is rapidly oscillating when the integer $\vert j\vert$ is large. From the viewpoint of engineering one says that the wave train $\exp \{ i(t-2\pi \ell / \varepsilon ) (j+\frac{1}{2}) \varepsilon \}$ is getting modulated by the `sinc' function. The resultant wave train amplitude has its maximum at $t=\frac{2\pi \ell}{\varepsilon}$ . From the viewpoint of physics one says that the wave trains $\{\frac{e^{i\omega t}}{\sqrt{2\pi}}\colon j\varepsilon <\omega <(j+1)\varepsilon\}$ comprising the wave packet exhibit a beating phenomenon with the result that they interfere constructively at $t=\frac{2\pi \ell}{\varepsilon}$ . From the viewpoint of mathematics one observes that the integral has a maximum value when the integrand does not oscillate, i.e. when $t=\frac{2\pi \ell}{\varepsilon}$ .

Second, we observe that the spacing between successive zeroes is $\Delta t= \frac{2\pi}{\varepsilon}$ . They are located at

$$t=\frac{2\pi}{\varepsilon}k\quad k=0,\pm 1,\cdots~~\textrm{but}~k\not= \ell~.$$

At $t=\frac{2\pi}{\varepsilon}\ell$ the wave packet $P_{j\ell }(t)$ has maximum modulus $\sqrt{\frac{\varepsilon}{2\pi}}$ . These two properties are summarized by the sifting property of $P_{j\ell }(t)$ :

$$P_{j\ell}\left( t= \frac{2\pi}{\varepsilon} k\right) = \sqrt{\frac{\varepsilon}{2\pi}}\delta_{k\ell}\,.$$ (262)

Consequently, the real and imaginary parts of the wave packets have profiles as depicted in Figure 2.12.

Figure: Real parts of the three wave packets $\sqrt{\pi}P_{j\ell}(2\pi t), ~\ell= 0,1,2$ as given by Eq.(2.61). The $t$ -axis is in units of $2\pi$ , i.e. it is expressed in periods of some standard clock. The width of the Fourier window is taken to be $\varepsilon =2$ . The mean frequency of all three wave packets is $(j+\frac {1}{2})\varepsilon =(2+\frac {1}{2})2=5$ oscillations per period.

Third, it has mean frequency $(j+\frac {1}{2})\varepsilon$ . Its mean position along the time axis is $\frac{2\pi}{\varepsilon}\ell$ . Its frequency spread is the width of its frequency window in the Fourier domain

$$\Delta\omega = \varepsilon\,.$$

Its temporal spread,

$$\Delta t = \frac{2\pi}{\varepsilon}\, \quad ,$$

is its half width centered around its maximum, which is located at $t=\frac{2\pi \ell}{\varepsilon}$ . Consequently, the frequency spread times the temporal spread of each wave packet is

$$\Delta\omega\Delta t = 2\pi\,,$$

which is never zero. Thus, the only way one can increase the temporal resolution $(\Delta t\to 0)$ is at the expense of the frequency resolution, i.e., by increasing $(\Delta\omega\to\infty )$ the frequency bandwidth of each wave packet. Conversely, the only way to increase the frequency resolution is to increase the width of the wave packet.

The last property is expressed by the following exercise:

Exercise 24.1 (ORTHONORMALITY AND COMLETENESS)
Consider the set of functions ("wave packets")

$$\Big\lbrace P_{jl} (t) = \frac{1}{ {\sqrt {\varepsilon}}} \int\l... ...{c} j = 0,\pm 1,\pm 2,\cdots\\ l = 0,\pm 1,\pm 2,\cdots \end{array}\Big\rbrace$$

where $\varepsilon$ is a fixed positive constant.

(a)

SHOW that these wave packets are orthonormal:

$${\rm i.e.} \quad\int\limits^\infty_{-\infty} P_{jl} (t) \bar P_{j'l'} (t)\,dt = \delta_{jj'} \delta_{ll'}$$

(b)

SHOW that these wave packets form a complete set:

$${\rm i.e.} \quad\sum\limits^\infty_{j= -\infty}\sum\limits^\infty... ...ver 2\pi}\int_{-\infty}^\infty e^{i\omega(t-t')}~d\omega \equiv \delta (t - t')$$ (263)

Note that the expression for a periodic train of delta functions, Poisson's sum formula, Eq.(2.20) on page , may be helpful here.

The completeness relation, Eq.(2.63) is equivalent to the statement that any square integrable function $f(t)\in L^2(-\infty ,\infty )$ can be represented as a superposition of wave packets, namely

$$f(t)=\sum^\infty_{j=-\infty}~\sum^\infty_{\ell =-\infty}\alpha_{j\ell} P_{j\ell} (t)\,,~\quad~ -\infty <t<\infty$$

where

$$\alpha_{j\ell} = \int^\infty_{-\infty}\overline{P}_{j\ell} (t')f(t')\,dt'$$

are the expansion coefficients.